# Lack of intuition for distribution function in micro and macro state description

I am a mathematician who is trying to understand statistical mechanics / thermodynamics. I need a hint wrt the interpretation / meaning of the distribution function. Currently I seem to have a basic misunderstanding which is a show stopper for further progress. My current understanding is as follows:

Micro state: I have a large number of particles ($$n$$) and a phase space $${\cal Q}$$ of the corresponding Hamiltonian system, i.e. $${\Bbb R}^{6n}$$. I consider points in this phase space and trajectories through it.

Macro state: I want to concentrate on more essential aspects of the system, such as descriptions in terms of energy, pressure, temperature etc. So, one macro state $$m$$ corresponds to a possibly very large set $$M$$ of micro states. In principle, given $$m$$, I could find $$M$$.

My Problem: Numerous texts now move ahead and introduce a distribution function or probability density on the set of micro states. I do not understand why I should be interested in considering a distribution function on the micro states.

Suppose, I have macro state $$m$$. So a micro state description could, in my understanding, already be given by the set $$M \subset {\cal Q}$$ of micro states. By the indifference principle I could assume that all of them are equally probable, which would give me some sort of distribution function / probability density. However, I read the texts in such a manner, that there could be several different distribution functions / probability densities. I want to understand which additional physical insight / intuition is modeled by the additional structure provided when moving from the set $$M$$ to a distribution function / probability density on the set.

The reason for introducing a probability distribution on the microstates is rooted in the ensemble formulation of statistical mechanics. Its goal is to allow to evaluate average values as averages over the set of microstates instead as time averages. The reason, from the theoretical perspective, is evident. A time average would imply to solve the equation of motion for a large number of interacting particles, while an ensemble average completely circumvents the step connected to the dynamical description. Even nowadays, when both approaches are possible with numerical sumulation tools (Molecular Dynamics vs Monte Carlo methods) ensemble averages for some problem may provide a significant advantage.

• I now understand the theoretical goal better but I still do not know the significance of the difference in two macro descriptions D1 and D2, where the sets of micro states are equal but the probability density / distribution function is different. What does the distribution on the sets of micro states "know" that the set of micro states does not yet know? What is the added modeling benefit? – Nobody-Knows-I-am-a-Dog Mar 27 '19 at 13:03
• @On-The-Internet-Nobody-Knows Macrostate definition implicitly must affect the robability of microstates. For example, if the total energy is fized, every microstate with a different energy must have zero probability. If the system has to exchange energy with a thermostat, in principle all microstates ar possible, but with different probability. – GiorgioP Mar 27 '19 at 17:13